Essentially Every Unimodular Matrix Defines an Expander

Abstract

We generalize the construction of Gabber and Galil to essentially every unimodular matrix in SL2(Z). It is shown that every parabolic or hyperbolic fractional linear transformation explicitly defines an expander of bounded degree and constant expansion. Thus all but a vanishingly small fraction of unimodular matrices define expanders.

Research supported in part by NSF grant CCR-9820806 and by a Guggenheim Fellowship.